Engineering with Computers

, Volume 30, Issue 2, pp 237–252 | Cite as

Geometric reasoning in sketch-based volumetric decomposition framework for hexahedral meshing

  • Jean Hsiang-Chun Lu
  • Inho Song
  • William Roshan Quadros
  • Kenji Shimada
Original Article


This paper presents a sketch-based volumetric decomposition framework using geometric reasoning to assist in hex meshing. The sketch-based user interface makes the framework user-friendly and intuitive, and the geometric reasoning engine makes the framework smarter and improves the usability. The system first generates a database that contains both the B-rep and 3D medial object to capture the exterior and interior of the input model, respectively. Next, the geometric reasoning process determines sweeping direction and two types of sweepable regions and provides visual aids to assist the user in developing decomposition solutions. The user conducts decomposition via the sketch-based user interface, which understands the user’s intent through freehand stroke inputs for smart decomposition. Imprint and merge operations are then performed on the decomposed model before passing it to the sweeping algorithm to create hex meshes. The proposed framework has been tested on industrial models.


3D medial object Geometric reasoning Hexahedral meshing Sketch-based decomposition 



The authors would like to thank Dr. Geoffrey Butlin, Mr. Henry Bucklow, Mr. Robin Fairey, Mr. Mark Gammon, Mr. Mike Field, and Mr. John Lamont for assisting with the medial related work in CADFIX.


  1. 1.
    Steven EB, Ernest P, Karl M, Brett C, Greg S (1995) A comparison of all-hexahedral and all-tetrahedral finite element meshes for elastic and elasto-plastic analysis. In: Proceedings of the 4th International Meshing Roundtable, pp 179–191Google Scholar
  2. 2.
    Blum H (1967) A transformation for extracting new descriptors of shape. In: Walthem-Dunn (ed) Models for the perception of speech and visual form. MIT Press, Cambridge, MA, pp 362–380Google Scholar
  3. 3.
    Chong CS, Senthil Kumar A, Lee KH (2004) Automatic solid decomposition and reduction for non-manifold geometric model generation. Comput Aided Des 36(13):1357–1369CrossRefGoogle Scholar
  4. 4.
    Cifuentes AO, Kalbag A (1992) A performance study of tetrahedral and hexahedral elements in 3-d finite element structural analysis. Finite Elem Anal Des 12:313–318CrossRefGoogle Scholar
  5. 5.
    Donaghy RJ, Armstrong CG, Price MA (2000) Dimensional reduction of surface models for analysis. Eng Comput 16:24–35CrossRefGoogle Scholar
  6. 6.
    Folwell NT, Mitchell SA (1998) Reliable whisker weaving via curve contraction. In: Proceedings of the 7th International Meshing Roundtable, pp 365–378Google Scholar
  7. 7.
    Hardwick M (2005) In DART system analysis presented to simulation sciences seminarGoogle Scholar
  8. 8.
    Igarashi T, Matsuoka S, Tanaka H (1999) Teddy: a sketching interface for 3D freeform design. In: Proceeding of the 26th annual conference on computer graphics and interactive, pp 409–416Google Scholar
  9. 9.
    ITI TranscenData (2013) CAD Translation-CADFix. In:
  10. 10.
    Kara LB, Shimada K (2006) Construction and modification of 3D geometry using a sketch-based interface. In: Proceeding of the EUROGRAPHICS Workshop on sketch-based interfaces and modeling, pp 59–66Google Scholar
  11. 11.
    Li TS, McKeag RM, Armstrong CG (1995) Hexahedral meshing using midpoint subdivision and integer programming. Comput Methods Appl Mech Eng 124(1-2):171–193CrossRefGoogle Scholar
  12. 12.
    Lu JHC, Song I, Quadros WR, Shimada K (2010) Pen-based user interface for geometric decomposition for hexahedral mesh generation. In: Proceedings of the 19th International Meshing Roundtable, pp 263–278Google Scholar
  13. 13.
    Lu Y, Gadh R, Tautges TJ (2001) Feature based hex meshing methodology: feature recognition and volume decomposition. Comput Aided Des 33(3):221–232CrossRefGoogle Scholar
  14. 14.
    Luo X-J, Shephard MS, Yin L-Z, OBara RM, Nastasi R, Beall MW (2010) Construction of near optimal meshes for 3d curved domains with thin sections and singularities for p-version method. Comput Methods Appl Mech Eng 26:215–229Google Scholar
  15. 15.
    Makem JE, Armstrong CG, Robinson TT (2012) Automatic decomposition and efficient semi-structured meshing of complex solids. In: Proceeding of the 20th International Meshing Roundtable. Springer, Berlin, Heidelberg, pp 199–215Google Scholar
  16. 16.
    Masry M, Kang D, Lipson H (2005) A freehand sketching interface for progressive construction of 3D objects. Comput Gr 29(4):563–575CrossRefGoogle Scholar
  17. 17.
    Owen SJ, Clark B, Melander DJ, Brewer ML, Shepherd J, Merkley KG, Ernst C, Morris R (2007) An immersive topology environment for meshing. In: Proceeding of the 16th International Meshing Roundtable, Sandia National Laboratories, pp 553–578Google Scholar
  18. 18.
    Pointwise Inc (2011) Multi-block grids for axial turbines. In:
  19. 19.
    Price MA, Armstrong CG (1997) Hexahedral mesh generation by medial surface subdivision: part II. Solids with flat and concave edges. Int J Numer Methods Eng 40(1):111–136CrossRefGoogle Scholar
  20. 20.
    Price MA, Armstrong CG, Sabin MA (1995) Hexahedral mesh generation by medial surface subdivision: part I. Solids with convex edges. Int J Numer Methods Eng 38(19):3335–3359CrossRefMATHGoogle Scholar
  21. 21.
    Quadros WR, Ramaswami K, Prinz FB, Gurumoorthy B (2004) LayTracks: a new approach to automated geometry adaptive quadrilateral mesh generaton using medial axis transform. Int J Numer Meth Eng 61:209–237CrossRefMATHGoogle Scholar
  22. 22.
    Sampl P (2000) Semi-structured mesh generation based on medial axis. In: Proceeding of the 9th International Meshing Roundtable, pp 21–32Google Scholar
  23. 23.
    Sandia National Laboratories (2013) Cubit: Geometry and meshing toolkit. In:
  24. 24.
    Schneiders R (1995) Automatic generation of hexahedral finite element meshes. In: Proceedings of the 4th International Meshing Roundtable, pp 103–114Google Scholar
  25. 25.
    Schneiders R (1996) A grid-based algorithm for the generation of hexahedral element meshes. Eng Comput 12:168–177CrossRefGoogle Scholar
  26. 26.
    Schoof L, Yarberry V (1995) Exodus II a finite element data model. SAND92-2137, Sandia National LaboratoriesGoogle Scholar
  27. 27.
    Sheffer A, Etzion M, Bercovier M (1999) Hexahedral mesh generation using the embedded voronoi graph. In: Proceedings of the 7th International Meshing Roundtable, pp 347–364Google Scholar
  28. 28.
    Shepherd JF, Johnson CR (2008) Hexahedral mesh generation constraints. Eng Comput 24(3):195–213CrossRefGoogle Scholar
  29. 29.
    Shih BY, Sakurai H (1996) Automated hexahedral mesh generation by swept volume decomposition and recomposition. In: Proceeding of the 5th International Meshing Roundtable, pp 273–280Google Scholar
  30. 30.
    Shih BY, Sakurai H (1997) Shape recognition and shape-specific meshing for generating all hexahedral meshes. In: Proceeding of the 6th International Meshing Roundtable, pp 197–209Google Scholar
  31. 31.
    Tam T, Armstrong CG (1991) 2D finite element mesh generation by medial axis subdivision. Adv Eng Softw 13(5–6):313–324MATHGoogle Scholar
  32. 32.
    Tautges TJ, Blacker T, Mitchell SA (1996) The whisker weaving algorithm: a connectivity-based method for constructing all-hexahedral finite element meshes. J Numer Methods Eng 39:3327–3349Google Scholar
  33. 33.
    Timothy JT (2000) The common geometry module (CGM): a generic, extensible geometry interface. In: Proceeding of the 9th International Meshing Roundtable, Sandia National Laboratories, pp 337–348Google Scholar
  34. 34.
    Varley PAC, Suzuki H, Mitani J, Martin RR (2000) Interpretation of Single Sketch Input for Mesh and Solid Models. Int J Shape Model 6:207–240Google Scholar
  35. 35.
    White D, Mingwu L, Benzley SE, Sjaardema GD (1995) Automated hexahedral mesh generation by virtual decomposition. In: Proceeding of the 4th International Meshing Roundtable, Sandia National Laboratories, pp 165–176Google Scholar
  36. 36.
    Yamakawa S, Gentilini I, Shimada K (2011) Subdivision templates for converting a non-conformal hex-dominant mesh to a conformal hex-dominant mesh without pyramid elements. Eng Comput 27:51–65CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Jean Hsiang-Chun Lu
    • 1
  • Inho Song
    • 1
  • William Roshan Quadros
    • 2
  • Kenji Shimada
    • 1
  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.Sandia National LaboratoriesAlbuquerqueUSA

Personalised recommendations